Question: Simplify the following expression and state the condition under which the simplification is valid. $y = \dfrac{3k^3 - 42k^2 + 135k}{5k^2 + 20k - 225}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ y = \dfrac {3k(k^2 - 14k + 45)} {5(k^2 + 4k - 45)} $ $ y = \dfrac{3k}{5} \cdot \dfrac{k^2 - 14k + 45}{k^2 + 4k - 45} $ Next factor the numerator and denominator. $ y = \dfrac{3k}{5} \cdot \dfrac{(k - 5)(k - 9)}{(k - 5)(k + 9)}$ Assuming $k \neq 5$ , we can cancel the $k - 5$ $ y = \dfrac{3k}{5} \cdot \dfrac{k - 9}{k + 9}$ Therefore: $ y = \dfrac{ 3k(k - 9)}{ 5(k + 9)}$, $k \neq 5$